Theorem 5.3 [1]}
If \(X:G\rightarrow TG\)  is an Euler-like multiplicative vector field for \((G,G_\mathcal {O})\)  then the tubular neighborhood embedding \(\phi :\nu (G_\mathcal {O})\rightarrow G\)  defined by \(X\)  determines a Lie groupoid tubular neighborhood of \(G_{\mathcal {O}}\rightrightarrows \mathcal {O}\) . In particular, if \(G\rightrightarrows M\)  is proper then there always exists such an Euler-like multiplicative vector field for \((G,G_\mathcal {O})\) .

